\(\int \sqrt {b x^3} \, dx\) [77]

Optimal result

Integrand size = 9, antiderivative size = 14 \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} x \sqrt {b x^3} \]

[Out]

2/5*x*(b*x^3)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} x \sqrt {b x^3} \]

[In]

Int[Sqrt[b*x^3],x]

[Out]

(2*x*Sqrt[b*x^3])/5

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b x^3} \int x^{3/2} \, dx}{x^{3/2}} \\ & = \frac {2}{5} x \sqrt {b x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} x \sqrt {b x^3} \]

[In]

Integrate[Sqrt[b*x^3],x]

[Out]

(2*x*Sqrt[b*x^3])/5

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(\frac {2 x \sqrt {b x}}{3}\)	\(9\)
gosper	\(\frac {2 x \sqrt {b \,x^{3}}}{5}\)	\(11\)
default	\(\frac {2 x \sqrt {b \,x^{3}}}{5}\)	\(11\)
trager	\(\frac {2 x \sqrt {b \,x^{3}}}{5}\)	\(11\)
risch	\(\frac {2 x \sqrt {b \,x^{3}}}{5}\)	\(11\)

[In]

int((b*x^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*x*(b*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} \, \sqrt {b x^{3}} x \]

[In]

integrate((b*x^3)^(1/2),x, algorithm="fricas")

[Out]

2/5*sqrt(b*x^3)*x

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \sqrt {b x^3} \, dx=\frac {2 x \sqrt {b x^{3}}}{5} \]

[In]

integrate((b*x**3)**(1/2),x)

[Out]

2*x*sqrt(b*x**3)/5

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} \, \sqrt {b x^{3}} x \]

[In]

integrate((b*x^3)^(1/2),x, algorithm="maxima")

[Out]

2/5*sqrt(b*x^3)*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} \, \sqrt {b x} x^{2} \mathrm {sgn}\left (x\right ) \]

[In]

integrate((b*x^3)^(1/2),x, algorithm="giac")

[Out]

2/5*sqrt(b*x)*x^2*sgn(x)

Mupad [B] (verification not implemented)

Time = 5.81 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^3} \, dx=\frac {2\,x\,\sqrt {b\,x^3}}{5} \]

[In]

int((b*x^3)^(1/2),x)

[Out]

(2*x*(b*x^3)^(1/2))/5